The subject matter disclosed herein relates generally to imaging systems and more particularly, to methods and systems for performing model-based iterative reconstruction for a dual-energy computed tomography (CT) imaging system.
CT imaging systems may be configured to perform dual energy scanning. Dual energy scanning may be used to obtain diagnostic CT images that enhance contrast separation within the image by utilizing two scans at different chromatic energy states. A number of techniques are known to achieve dual energy scanning, including acquiring two back-to-back scans sequentially in time where the scans require two rotations around the object in which the tube operates at, for instance, 80 kilovolt peak (kVp) and 140 kVp potentials. For example, the x-ray source may be operated at a first kVp setting to acquire a first set of data and then operated, at a second kVp to acquire a second set of data. Optionally, the x-ray source may be operated to continuously switch from the first kVp setting to the second kVp during the acquisition such that the resultant data acquired at the first kVp setting is interleaved with the data acquired at the second kVp setting.
Acquiring X-ray CT exposures at several distinct energy levels enables an operator to distinguish different types of basis material which is useful for disease diagnosis and security inspection. At least one known dual-energy CT reconstruction technique is utilized to reconstruct two density maps for the two basis materials. A cross-sectional attenuation map, emulating the image that would be obtained from a monochromatic acquisition at any given energy, may then be computed as a linear combination of the two material density maps. One known method to perform dual-energy reconstruction initially transforms the low- and high-energy photon counts into quantities that are proportional to the integral of the material density for two basis materials. A material-decomposed sinogram is then filtered backprojected (FBP) to reconstruct the material density maps in image space. The transformation from photon counts to integral projections in the material basis pairs is typically performed using a material-decomposition function which may be experimentally measured through a scanner calibration procedure. However, the processes of applying the material-decomposition function changes the statistics of the measured data, which results for FBP in reconstructions that have statistically correlated noise properties.
Statistical iterative reconstruction approaches such as model-based iterative reconstruction (MBIR) have been shown to have advantages in the reconstruction of conventional (single-spectrum) CT data. More specifically, using models of the scanner geometry, data noise, and the statistics of the projection data for the image to be reconstructed, facilitates both reducing noise and improving resolution in the reconstruction. Statistical algorithms that explore the interactions between measurement statistics and dual-energy reconstruction have been proposed. For example, one algorithm utilizes a penalized likelihood approach for a poly-energetic model based on Poisson statistics to correct for beam-hardening artifacts. However, the reconstruction is still performed from single-energy data. Another algorithm utilizes a penalized weighted least-squares (PWLS) method with image-domain regularization under a monochromatic assumption for source spectra.
More generally, a concern of dual or multi-energy CT is how to reconstruct an image of material densities, x, from dual or multi-energetic sinogram measurements, y. For example, for a dual energy system with two spectral measurements, the dual-energy sinogram is given by yi,k where i=1, . . . M indexes the projections and k=1, 2 indexes the two energies. In this case, each voxel of the reconstruction represents the densities of two basis materials, xj,k, where j=1, . . . N indexes the voxel and k indexes one of the two basis materials. The basis materials may be, for example, water and iodine.
One known method of iteratively reconstructing dual energy CT data performs direct reconstruction of x from the multi-spectral measurements y. In this method, the reconstruction is produced through minimization of a cost function having the general form:x^=arg min {D(y; x)+S(x)}  Equation 1
where D(y; x) is a function which models the likelihood of observing the multi-energy measurements of y given the hypothesized reconstruction of x, and S(x) is a stabilizing function which regularizes the problem by assigning a larger cost to density images, x, that are unlikely to be correct. The disadvantage of this method is that the function D(y; x) is generally very complex and computationally difficult to model due to the nonlinear relationship between x and y.
Optionally, y may be transformed into a material-decomposed sinogram using a nonlinear transformation operator h−1. Using the nonlinear transformation operator h−1, the material decomposed sinogram may be expressed as: [^p_{i,1},p^_{i,2}]=h−1(y_{i,1},y_{i,2}), where for p^{i,k}, i=1, . . . M indexes the projections and k indexes one of the two basis materials. The material-decomposed sinogram is linearly related to x through the projection integrals. More specifically, the resulting decomposed sinogram corresponds to a set of material projections for each corresponding material component of x. Accordingly, when the noise is relatively small: p^=Ax. This is an advantage over direct reconstruction via Equation 1 since the standard framework for MBIR that assumes the linearity of the relationship between p and x can be applied without dealing with the complexity of D(y; x) explicitly.
In practice, the form of the material decomposition operator, h−1, may be estimated from system calibrations and the known physical behavior of various materials. For example, in one known method, the material decomposition sinogram is used to simplify the formulation of the iterative reconstruction problem. In this method the problem is formulated as:
                              x          ^                =                  arg          ⁢                                          ⁢                                    min              x                        ⁢                          {                                                                                          ∑                                              k                        =                        1                                            K                                        ⁢                                                                  D                        k                                            ⁡                                              (                                                                                                            p                              ^                                                                                      *                              k                                                                                ;                                                      x                                                          *                              k                                                                                                      )                                                                              +                                      (                                          S                      ⁡                                              (                        x                        )                                                              }                                                  ,                                                                Equation      
where K=2 and {circumflex over (p)}*k denotes the estimated material decomposition sinogram for all i indexes for the k=1 and k=2 basis materials, and x*k denotes the material density reconstruction for all j indexes for the k=1 and k=2 basis materials. This method is computationally more direct than the method using minimization of a cost function described by Equation 1, but the decoupled terms of Equation 2 do not fully account for the dependencies between sinogram entries that are caused by the application of the material decomposition operator.
In another known method, the material decomposed sinograms are directly estimated using an iterative sinogram restoration technique. The sinogram restoration formulation then uses a Penalized Weighted Least-Square (PWLS) framework to estimate the material decomposed projections, wherein the weights are computed to approximate the inverse covariance of the decomposed sinograms. However, image reconstruction is still performed with FBP. Another proposed method of reconstruction provides a two-step PWLS approach, wherein a first PWLS estimates the material sinograms from the dual-energy data and a second PWLS reconstructs iteratively the images from the estimated sinogram using a diagonal weighting matrix that does not take into account the statistical correlation between the material sinograms. New joint regularization in the material component densities was also introduced to preserve edges at the same spatial locations. Additionally, the dual-energy problem may be formulated as a double minimization 1-divergence problem. However, the double minimization 1-divergence approach does not specifically account for the correlation of material sinograms in the cost function.
Accordingly, known methods directed to model-based dual-energy reconstruction do not fully model the statistical dependencies in the material-decomposed data. More specifically, the known methods treat the two (or more) material-decomposed sinograms as statistically independent, or equivalently, these methods minimize a cost function with a separate term for each component of the material-decomposed sinograms.